# Search Results

## Abstract

We prove that for any given *c*, 1 < *c* < 17/11, almost all natural numbers are representable in the form [*x*
^{c}] + [*p*
^{c}], where *x* is a natural number and *p* is a prime.

Let *P*
_{
r
} denote an almost-prime with at most *r* prime factors, counted according to multiplicity, and let *E*
_{3}(*N*) denote the number of natural numbers not exceeding *N* that are congruent to 4 modulo 24 yet cannot be represented as the sum of three squares of primes and the square of one *P*
_{5}. Then we have *E*
_{3}(*N*)≪log^{1053}
*N*. This result constitutes an improvement upon that of D. I. Tolev, who obtained the same bound, but with *P*
_{11} in place of *P*
_{5}.

## Abstract

We sharpen Hua’s result by proving that each sufficiently large odd integer *N* can be written as

*p*

_{j}are primes. This result is as good as what was previously derived from the Generalized Riemann Hypothesis.

## Abstract

*N*satisfying some necessary congruence conditions are the sum of

*j*almost equal prime cubes with

*j*= 5; 6; 7; 8, i.e.,

*N*=

*p*

_{1}

^{3}+ ... +

*p*

_{ j }

^{3}with |

*p*

_{ i }− (

*N/j*)

^{1/3}| ≦

*i*≦

*j*), for

*δ*

_{ j }= 1/45; 1/30; 1/25; 2/45, respectively.

## Abstract

Let *p*
_{
i
} be prime numbers. In this paper, it is proved that for any integer *k*≧5, with at most exceptions, all positive even integers up to *N* can be expressed in the form . This improves the result for some *c*>0 due to Lu and Shan [12], and it is a generalization for a series of results of Ren and Tsang [15], [16] and Bauer [1–4] for the problem in the form . This method can also be used for some other similar forms.

## Abstract

*N*congruent to 5 modulo 24 can be written in the form

*p*

_{1}≦

## Abstract

We show that if *A* is a subset of {1, …, *n*} which has no pair of elements whose difference is equal to *p* − 1 with *p* a prime number, then the size of *A* is *O*(*n*(log log *n*)^{−c(log log log log log n)}) for some absolute *c* > 0.

## Abstract

We show that for every fixed *A* > 0 and *θ* > 0 there is a *ϑ* = *ϑ*(*A, θ*) > 0 with the following property. Let *n* be odd and sufficiently large, and let *Q*
_{1} = *Q*
_{2}:= *n*
^{1/2}(log *n*)^{−ϑ
} and *Q*
_{3}:= (log *n*)^{
θ
}. Then for all *q*
_{3} ≦ *Q*
_{3}, all reduced residues *a*
_{3} mod *q*
_{3}, almost all *q*
_{2} ≦ *Q*
_{2}, all admissible residues *a*
_{2} mod *q*
_{2}, almost all *q*
_{1} ≦ *Q*
_{1} and all admissible residues *a*
_{1} mod *q*
_{1}, there exists a representation *n* = *p*
_{1} + *p*
_{2} + *p*
_{3} with primes *p*
_{
i
} ≡ *a*
_{
i
} (*q*
_{
i
}), *i* = 1, 2, 3.

## Abstract

Let denote the set {*n*∣2|*n*, ∀ *p*>2 with *p*−1|*k*}. We prove that when , almost all integers can be represented as the sum of a prime and a *k*-th power of prime for *k*≧3. Moreover, when , almost all integers *n*∊(*X*,*X*+*H*] can be represented as the sum of a prime and a *k*-th power of integer for *k*≧3.

## Abstract

It is proved that every sufficiently large even integer is a sum of one prime, one square of prime, two cubes of primes and 161 powers of 2.